First of all is clear that KL is sufficient for order-coherence as the optimal valid belief update is then equal to the Gibbs posterior \(\psi ^{(q)}\)  (see below) and we have seen this is order-coherent under the conditions of Theorem REF . In order to show KL is necessary it is enough to give an example where the KL divergence is the only \(g\) -divergence giving coherence, so [1]} take a parameter space with just two states, \(\Omega =\lbrace \xi _1,\xi _2\rbrace \)  say. Let \(O^{(1)}=(Y^{(1)},Z^{(1)})\) , \(O^{(2)}=(Y^{(2)},Z^{(2)})\)  and
\({\tilde{q}}_1(\xi )\propto \exp (-l(\xi ;O^{(1)},\pi _0))\pi _0(\xi )\) 
